Tutorial 7

ELE3410 Random Processes and DSP THE CHINESE UNIVERSITY OF HONG KONG Department of Electronic Engineering ELE3410 Random Processes and DSP Tutorial 6 Power Spectral Density 2D Gaussian and n-D Gaussian 1. Power Spectral Density What is Power Spectral Density? Parseval’s Theorem (for aperiodic signal): ∞ ∞2 2y (t) dt = |Y ( f ) | df ∫ ∫− ∞ − ∞For ergodic random process, its average power: T1 2lim x (t)dt∫T → ∞ 2T −T∞1 2= lim | X ( f ) | df∫T → ∞ 2T − ∞∞= G( f ) df∫− ∞1 2 G( f ) ≡ lim | Χ( f ) |Then . T → ∞ 2TNote: (1)G(f) ≥ 0 ∀ f (2)G(f) = G(-f) G(f) power spectral density [watt/Hz] Tutorial 6 [1/5] ELE3410 Random Processes and DSP Power spectral density function(PSD) shows the strength of the variations (energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak. The unit of it is energy per frequency and you can obtain energy within a specific frequency range by integrating PSD within that frequency range. How to compute PSD? Computation of PSD is done by computing autocorrelation function and then transforming it using DFT. G(f) & R(τ) constitute a Fourier Transform pair: ∞ ∞1− j2 πf τ j2 πf τG( f ) = R( τ) ⋅e d τ & R( τ) = G( f ) ⋅e df ∫ ∫2 π− ∞ − ∞ Question 1 (Assignment 6) Find the power spectral density G( ω) corresponding to the autocorrelation function 1 for| τ | < T⎧ 0 .⎨R( τ) ...